Chapter 2: Equilibrium Thermodynamics and Kinetics

System: that portion of the universe we wish to study. It could be as simple as a beaker containing solution or as complex as the universe. A system can be open (exchanging matter and energy with its surroundings), closed (not exchanging matter with its surroundings), or isolated (exchanges neither matter nor energy with its surrounding.) Equilibrium Thermodynamics: predicts the concentrations (or more precisely, activities) of various species and phases if a reaction reaches equilibrium. Kinetics tells us how fast, or if, the reaction will reach equilibrium.

The First Law of Thermodynamics: (a.k.a. Conservation of Energy) Energy can neither be created nor destroyed, it can only be changed from one form to another. The change in internal energy of a system is equal to the heat added to the system minus the work done by the system. ΔE = q – w Work decreases energy!

Internal energy (E): the energy associated with the random, disordered motion of molecules. Heat (q): energy in transit from a high temperature object to a lower temperature object. Work (w): forms of energy transfer which can be accounted for in terms of changes in the macroscopic physical variables of the system. ΔE = q – w

Remember: ΔE = q – w Now to the math… If the work done by or on a system causes a change in volume at constant pressure, then the equation for ΔE becomes: ΔE = q – PΔV. Enthalpy(H) is equal to the heat flow when processes occur at constant pressure and the only work done is pressure- volume work. dH = dE + PdV So…how does enthalpy influence a reaction?

Exothermic reactions: reactions that release heat energy (enthalpy is negative for the reaction). Reactants → products + heat CH 4 + 2O 2 → CO 2 + 2H 2 O + heat Endothermic reactions: reactions that use heat energy (enthalpy is positive for the reaction). Reactants + heat → products Ba(OH) 2 ·8H 2 O (s ) + 2NH 4 SCN (s ) → Ba(SCN) 2(s ) + 10H 2 O (l ) + 2NH 3(g ) Heat of formation: enthalpy change that occurs when a compound is formed from its elements at a specific (standard) temperature and pressure. The heat of formation for the most stable form of an element is arbitrarily set equal to zero.

Second law of thermodynamics: for any spontaneous process, the process always proceeds in the direction of increasing disorder—entropy. Heat energy

Another way to look at entropy is that during any spontaneous process, there is a decrease in the amount of usable energy. Consider the combustion of coal, an ordered complex organic molecule is through the process of combustion, broken down into ash, CO 2, H 2 O, SO x & NO x. Resulting in a dramatic decrease in the amount of usable energy.

Mathematically, entropy (S) is described in the following equation.  S = q/T Where  S is the change in entropy and T is the temperature in Kelvin. When the entropy equation is combined with the enthalpy equation and only pressure-volume work is considered, then: dH = TdS + VdP

Chemical Equilibrium Chemical equilibrium applies to reactions that can occur in both directions. In a reaction such as: CH 4(g) + H 2 O (g) CO (g) + 3H 2(g) Many chemical reactions are reversible. The products formed react to give back the original reactants, even as the reactants are forming more products. After some time, both the forward and reverse reactions will be going on at the same rate. When this occurs, the reaction is said to have reached equilibrium.

A system at equilibrium is in a state of minimum energy. This energy can be measured as: Gibbs free energy when the reaction occurs at constant temperature and pressure. For a system at constant T & P. Gibbs free energy can be written as: G = H – TS Where H is enthalpy(kJ mol -1 ), S is entropy(J mol -1 K -1 ) and T is temperature (K)

For changes that occur at constant T & P the expression for Gibbs free energy becomes:  G =  H – T  S If  G is (-), the process occurs spontaneously. If  G = 0, the process is at equilibrium If  G is (+), the reaction does not occur spontaneously To determine  G R º (for the entire reaction) you must subtract the  G of the reactants from the  G of the products.  G Reaction =  G Products –  G Reactants See appendix II in the back of the book (page 474). So…  G R =  H R - T  S R

Equilibrium Constant To determine the amount of each compound that will be present at equilibrium you must know the equilibrium constant. To determine the equilibrium constant you must consider the generic equation: aA + bB cC + dD The upper case letters are the molar concentrations of the reactants and products. The lower case letters are the coefficients that balance the equation. Use the following equation to determine the equilibrium constant (K c also written K eq ).

For example, determining the equilibrium constant of the following equation can be accomplished by using the Kc equation. Using the following equation, calculate the equilibrium constant. N 2(g) + 3H 2(g) 2NH 3(g) A one-liter vessel contains 1.60 moles NH 3,.800 moles N 2, and 1.20 moles of H 2. What is the equilibrium constant? 1.85

Incorporating Gibbs free energy with the equilibrium constant (under standard conditions) yields the following equation: logK eq = -  G R / Where is a combination of the gas constant (R) and standard temperature of 25°C. Example 2-1 Calculate the solubility product for gypsum at 25°C

The dissolution of Gypsum is written as follows: CaSO 4 ∙2H 2 O(gypsum) ↔ Ca 2+ + SO H 2 O The equilibrium constant is equal to the solubility constant. K eq = K sp [Ca 2+ ][SO 4 2- ][H 2 O] 2 K sp = ————————— [CaSO 4 ∙2H 2 O (gypsum) ] You can place the concentration of both water and gypsum at 1 as the solution is dilute and the gypsum is a solid in its standard state. K sp = [Ca 2+ ][SO 4 2- ] If you know the concentrations of calcium and sulfate, you can find the solubility constant with this equation. If not, you may use the free energies of formation from appendix II.

 G R =  G (products) –  G (reactants) For this problem:  G R = (-522.8) + (-744.0) + 2( ) – ( ) = kJ mol -1 logK sp = -  G R / = /5.708 = ∴ K sp =

This was ever so simple. Now, what if the activity of the species in the reaction is not equal to the concentrations in the solution?

Activity and Fugacity (for a gas): the apparent (or effective) concentration of a species as opposed the actual concentration. They are a measure of the departure of a system from ideal behavior and need to be taken into account even when dealing with relatively dilute solutions. Activity (or fugacity) is related to concentration through the activity coefficient (  i ).  i = a i / m i Where a i is the activity and m i is the actual concentration.

Activity coefficient (  ): measure of how a specific real system deviates from some reference system that is taken to be ideal. In an ideal solution, activity would equal concentration. The departure from ideal behavior is caused mainly by: Electrostatic interactions between charged ions. The formation of hydration shells around ions..

One factor that is important to understand is ionic strength ( I ). I = ½  m i z i 2 Where m i = the moles/liter of ion i and z i is the charge of ion i. **It is important to understand that I considers all of the ions in the solution in question. Debye-Hückel Model log  i = (-Az i 2 ( I ) 1/2 )/(1 + Ba i ( I ) 1/2 ) Where A and B are constants depending only on T and P and a i is the hydrated radius of a particular ion.

Example 2-4 Given the following river water chemistry, calculate the activity coefficient for Ca 2+ at 25°C, using both the Debye-Hückel ands the Truesdell-Jones equations. For our purposes, we will stick with the Debye-Hückel equation. River Water Concentration (mg L -1 ) Ca 2+ Mg 2+ Na + K+K+ Cl - SO 4 2- HCO 3 - SiO We must first find Calculate I based on all of the ions in the river water. Recall: I = ½  m i z i 2

Using the Debye Huckel equation for Ca 2+ : log  i = (-Az i 2 ( I ) 1/2 )/(1 + Ba i ( I ) 1/2 ) Log   Ca 2+ ) = ( )(2) 2 ( ) 1/2 )/(1 + (0.3289)(6.0)( ) 1/2 ) = ∴  (Ca 2+ ) = = 0.82 What does this imply about the activity of the calcium ion versus the actual concentration? Keep in mind that a i = (m i )(  i ) And please…… Do Not Fear the SPREADSHEET!!!!

Table 2-4 Appropriate Ranges of Ionic Strengths for Activity Coefficient Models ModelIonic Strength (mol/L) Debye-Hückel0 to 0.1 Davies0 to 0.6 Truesdell-Jones0 to 2 Specific ion interaction0 to 4 Pitzer0 to 6 We should be aware of the different types of activity coefficient models and their limitations.

The reality we face is that life is rarely ideal…. So how do we calculate free energies or solubility constants at temperatures other than 25 ⁰ C? Does temperature make a difference in solubility?

Free energies at temperatures other than 25°C If the deviations in temperature from 25°C are small (~15°C or less) we can make the assumption that  H R and  S R are constant. This brings us to the van’t Hoff equation: lnK t = lnK r + (  H R /R)(1/T r – 1/T t ) Where K t is the equilibrium constant at temperature t, K r is the equilibrium constant at 25°C, T t is the temperature t and T r is K. Of course,  H R is the enthalpy for the reaction, and R = X kJ/mol·K.

Example 2-3 page 34. Calculate the solubility product of gypsum at 40°C using the van’t Hoff equation. Recall that K sp at 25 ⁰C = for the dissolution of gypsum. From appendix II  H R = (-543.0)+( )+(2)( )-( ) = -1.08kJ mol -1 lnK t = ln( ) + [(-1.08/8.314X10 -3 )/((1/298.15)-(1/313.15))] = = e =

John Dalton studied the effect of gases in a mixture. He observed that the total pressure of a gas mixture was the sum of the partial pressure of each gas. P (total) = P 1 + P 2 + P P n Now, a quick review of partial pressure…

Henry’s Law: The relationship between the equilibrium constant and the solubility product is used to describe the activity of a dilute component as a function of concentration. Henry’s law relates the fugacity of a gas to its activity in solution. When a gas behaves ideally (1 atm and near surface temperatures) fugacity of a gas equals its partial pressure. Therefore, Henry’s law can be written: c i = K H P i Where c i is the concentration of the gaseous species i in solution, K H is Henry’s law constant in mol L -1 atm -1 (see table 2-1 pg 33), and P i is the partial pressure of gaseous species i.

Get to know this table. You will reference it more than once this semester……

Example 2-2 (page 33) Calculate the solubility of oxygen in water at 20°C. At sea level (a total atmospheric pressure of 1 bar; where one atm. = bar) the partial pressure of oxygen is 0.21bar. At 20°C, the Henry’s law constant for oxygen is 1.38 X mol/L·bar. C i = K H P i O 2(aq) = (1.38 X mol/L·bar)(0.21 bar) = 2.9 X mol/L Or (2.9 X mol/L)(32.0 gO 2 /mol) = 9.28 X g/L or 9.28 mg/L

Le Chatelier's principle If a change is imposed on a system at equilibrium, the position of the equilibrium will shift in a direction that tends to reduce the change. To review: CaSO 4 ·2H2O  Ca 2+ + SO H 2 O + heat If there is an increase in Ca 2+, what will happen? If the system’s temperature is decreased, what will happen? CaCO 3 + SiO 2  CaSiO 3 + CO 2(g) If the pressure on the system is increased, what will happen?

Le Chatelier's principle is HUGE! It is the 500 pound gorilla that cannot be ignored when you consider thermodynamic equilibrium. It is not, however, EVERYTHING….

A Measure of Disequilibrium Consider the dissolution of gypsum: CaSO 4 ·H 2 O  Ca 2+ + SO H 2 O K sp = [Ca 2+ ][SO 4 2- ] This specific reaction simplifies matters to the point that we can consider the activities of the products when determining K sp. We would call this the activity product [AP] or ion activity product if only charged species are involved [IAP]. If the [AP] = K sp, the system is in equilibrium,. If the [AP] < K sp, the solution is undersaturated with respect to gypsum. If the [AP] > than the K sp, the solution is supersaturated with respect to gypsum.

Example 2-6 In a particular solution [Ca 2+ ] = mol L -1 and [SO 4 2- ] = mol L -1. At 25 ⁰C, is the solution over- or under-saturated with respect to gypsum? IAP / K sp = ((10 -3 )(10 -2 ))/ = IAP is less than 1 so the solution is under-saturated with respect to gypsum.

Kinetics Equilibrium thermodynamics predicts the final state of a system. Kinetics tells us if the system will actually achieve this state within a reasonable time. It’s all relative…

Homogeneous reactions: involve one phase (gas, liquid or solid. Heterogeneous reactions: involve two or more phases. Reactions often occur in a series of steps. The slowest step determines the rate at which the reaction will proceed.

Systems can exist in several states: unstable, metastable and stable. Figure 2-1 in text

The dissolution of gypsum occurs in two main steps. 1 st : energy input is required to free the ions from the crystal structure. 2 nd : the ions must be diffused. If they are not, then the microenvironment surrounding the gypsum crystal becomes saturated and the reaction stops. CaSO 4 ·H 2 O  Ca 2+ + SO H 2 O Which ever these steps is the slowest, will determine the rate at which dissolution proceeds. A common environment for gypsum dissolution is in the presence of ground water. In this case, which step would determine the rate of dissolution?

Order of reactions Zeroth order: The reaction rate is independent of the concentration of the reactant (A) –rate of the reaction is constant. When the limiting reactant is completely consumed, the reaction abruptly stops. Units for k = mol L -1 sec -1 t 1/2 = 0.5A 0 /k First order: The reaction rate is dependent on the concentration of the reactant A  B. – The rate of the reaction is directly proportional to the concentration of one of the reactants. Units for k = sec -1 t 1/2 = 0.693/k Second order: The reaction rate is dependent on the concentration of the reactant 2A  B. –The rate of the reaction is directly proportional to the square of the concentration of one of the reactants. Units for k = L mol -1 sec -1 t 1/2 = - 1/kA 0 Note that the units for k depend upon the order of the reaction.

Changes in pH can also change the reaction rates of a solution. This would be indicated in the concentration of H + in the solution.

Figure 2-4 Illustrates the variability of dissolution rates with change in pH.

Consider the reaction rates & residence times of water for various reservoirs If t ½ < residence time. Then the process should reach equilibrium.

The Arrhenius Equation: relates the rate at which a reaction occurs to the temperature. k = A exp(-E a /RT) Or log k = log A – (E a /2.303RT) Where A is a experimentally derived factor, E a is the activation energy for the reaction, R is the ideal gas constant and T is temperature in Kelvin. Activation energy: the energy that must be overcome in order for a chemical reaction to occur. *Eby notes that a rough rule of thumb for activation energy is that it doubles with every 10ºC increase in temperature. Let us not forget the first equation in the chapter  E = q - w

Example 2-8 Consider one of the reactions in the carbonate system: CaCO 3calcite +H 2 CO 3(aq) → Ca HCO 3 - The rate constant for this reaction is 3.47 X s -1 and logk = The activation energy for the reaction us 41.85kJ mol -1. Determine A at 25 ⁰ C and then the rate constant for the reaction at 10 ⁰ C. log A = logk + E a /(2.303RT) = /((2.303)(8.314X10 -3 )(298.15)) = 2.87 logk = logA – E a /(2.303RT) = 2.87 – 41.85/(2.303)(8.314x10 -3 )(283.15) = If at 25 ⁰ C, k = and at 10 ⁰ C, k = then it can be said that the reaction rate has decreased by ~2.5 times as a result of the lower temperature.

Chapter 2 Homework: Please complete and turn in by Monday September 24 th From Chapter 2--Problem #s: 3, 4, 8, 19, 23, 25, 31, 36 As a reminder, there are no late assignments.